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Math Help - Closed in a dual space

  1. #1
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    Closed in a dual space

    Let X be a Banach space. We show p(X) is closed in X**, where p:X-->X** is defined by p(x)=T_x and T_x:X*-->F is defined by T_x(x*)=x*(x) (F is a field).

    I think I should pick a convergent sequence {x_n} in p(X) (x_n -->x)and show that x belongs to p(X). i.e. show there exists a w in X such that p(w)=x.
    But for some reason I am not getting the answer.
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    Quote Originally Posted by math8 View Post
    Let X be a Banach space. We show p(X) is closed in X**, where p:X-->X** is defined by p(x)=T_x and T_x:X*-->F is defined by T_x(x*)=x*(x) (F is a field).

    I think I should pick a convergent sequence {x_n} in p(X) (x_n -->x)and show that x belongs to p(X). i.e. show there exists a w in X such that p(w)=x.
    But for some reason I am not getting the answer.
    For x in X, \|p(x)\| = \sup\{\|T_x(x^*)\|:x^*\in X^*,\;\|x^*\|=1\} = \sup\{\|x^*(x)\|:\|x^*\|=1\} = \|x\| by the Hahn–Banach theorem. So the map p is isometric, and it follows easily from that that its range must be closed.
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